Question

Circle A has a radius of `2` and a center of `(3 ,7 )`. Circle B has a radius of `6` and a center of `(8 ,1 )`. If circle B is translated by `<-4 ,3 >`, does it overlap circle A? If not, what is the minimum distance between points on both circles?

Answer 1

The distance between the centers of the circle is less than the sum of the length of their radiuses. Thus the circles overlap and circle `O_B'` totally engulf circle `O_A`

`|3.16|< |2+6|`

Explanation:

Given : Circles A, B with centers and radiuses
`O_A(3,7;r=2); => (x-3)^2+(y-7)^2=4`
`O_B(8,1;r=6); => (x-8)^2+(y-1)^2=36`

Required :
If circle `O_B(8,1;r=6)` translates by (-4,3) to `=>O_(B')(x,y;r=6)`
Does `O_(B')(x,y;r=6)` overlap with `O_A(3,7;r=2)`

Solution Strategy:
a) Translate `O_B` by vector <-4,3> and find the new center
b) Calculate the distance from `O_A(3,7) <=> O_B'(x,y)=bar(AB')`
c) Is `|bar(AB')|<|r_A+r_B'|`
If the above is `"holds" => "it overlaps" if "False" => "it does not"`

a) Translation Matrix of center vector `<3,7>` by <-4,3> is
`T_(-4,3)[(8),(1)]= [(1-4,0),(0,1+3)] [(8),(1)]= [(8-4 ),(1+3)] =[(4),(4)]`
So the new center `O_(B')(x,y)=O_(B')(4,4)`

b) Use the distance formula to calculate `O_A(3,7) <=> O_B'(4,4)`
`bar(AB') = sqrt((3-4)^2+(7-4)^2) = sqrt((-1)^2+(3)^2) = sqrt(10)=3.16`

c) Is Is `|bar(AB')|<|r_A+r_B'|`
`|3.16|< |2+6|`
Since the distance from the center is less that the radius of `O_B'`, (about half of `O_B'`), the circles not only do they overlap but also `O_B'` totally engulf `O_A`